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Math Help - Suppose we have a right triangle with integer sides

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    Suppose we have a right triangle with integer sides

    Suppose we have a right triangle with integer sides (in some unit of measure). Prove
    (a) One of the legs has a length divisible by 3.
    (b) One of the three sides has length divisible by 5.
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  2. #2
    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by NikoBellic View Post
    Suppose we have a right triangle with integer sides (in some unit of measure). Prove
    (a) One of the legs has a length divisible by 3.
    We want to show either  m^2-n^2 or  2mn is divisible by  3 .

    WLOG if  m\equiv0\bmod{3} then we're done since then  2mn\equiv0\bmod{3} .

    So assume  m and  n are both not divisible by  3 .

    But then by flt we get that  m^2\equiv1\bmod{3} and  n^2\equiv1\bmod{3} , so  m^2-n^2\equiv0\bmod{3} .
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